# Probability/Statistics: $Z$- or $T$-test?

You have sampled the lengths of bird beaks from $50$ birds, and the mean beak length is $10 ~ \text{mm}$ with a variance of $3 ~ \text{mm}^{2}$. What is the probability that the $51^{\text{st}}$ beak is less than or equal to $12 ~ \text{mm}$ in length?

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Are you asking whether to use a z-test or a t-test, or how to compute the result of this question given a z-test or a t-test? –  TakeS Feb 20 '13 at 2:59
I wonder how to calculate the probability that the 51st beak is less than or equal to 12 mm in length? I think it could be as simple as Z=(Y- μ)/σ =(12-10)/1.73= 1.16 and the probability of 1.16 using the Z table is 0.1230...but I'm not 100% sure... thanks for any feedback! –  user63036 Feb 20 '13 at 3:41
Two things: (1) There's no way the probability of getting <= 1.16 is that low because it must logically be at least 0.5 (you may be reading something wrong). (2) This question doesn't make sense (to me, at least). You can't really use the z-scores if you don't know the true population mean and variance. It seems like this is more of a question where we are trying to compare the mean beak length of our current sample of 50 birds to the mean beak length of the entire population. –  TakeS Feb 20 '13 at 4:10